find-the-following-product-integral-1-x-dx-2-e-x-dx-

Question Number 110096 by mathdave last updated on 27/Aug/20

$${find}\:{the}\:{following}\:{product}\:{integral} \\ $$$$\:\:\:\:\:\:\:\:\left(\mathrm{1}\right)\:\:\:\:\int\left({x}\right)^{{dx}} \\ $$$$\:\:\:\:\:\:\:\:\:\left(\mathrm{2}\right)\:\:\:\:\int\left({e}^{{x}} \right)^{{dx}} \\ $$

Commented by Her_Majesty last updated on 27/Aug/20

if you use the same definition of product integral as me (1) (x/e)^x ×C (2) e^(x^2 /2) ×C

$${if}\:{you}\:{use}\:{the}\:{same}\:{definition}\:{of}\:{product} \\ $$$${integral}\:{as}\:{me} \\ $$$$\left(\mathrm{1}\right)\:\left({x}/{e}\right)^{{x}} ×{C} \\ $$$$\left(\mathrm{2}\right)\:{e}^{{x}^{\mathrm{2}} /\mathrm{2}} ×{C} \\ $$

Commented by mathdave last updated on 27/Aug/20

$${oh}\:{no}\:{that}\:{is}\:{not}\:{correct} \\ $$

Commented by Her_Majesty last updated on 27/Aug/20

$${then}\:{please}\:{post}\:{the}\:{definition} \\ $$

Commented by Her_Majesty last updated on 27/Aug/20

the definition I know is ∫f(x)^dx =e^(∫ln(f(x))dx) and as you should know there are many kinds of product integrals

$${the}\:{definition}\:{I}\:{know}\:{is} \\ $$$$\int{f}\left({x}\right)^{{dx}} ={e}^{\int{ln}\left({f}\left({x}\right)\right){dx}} \\ $$$${and}\:{as}\:{you}\:{should}\:{know}\:{there}\:{are}\:{many} \\ $$$${kinds}\:{of}\:{product}\:{integrals} \\ $$

Commented by mathdave last updated on 27/Aug/20